reserve E,V for set, G,G1,G2 for _Graph, c,c1,c2 for Cardinal, n for Nat;
reserve f for VColoring of G;
reserve g for EColoring of G;
reserve t for TColoring of G;

theorem Th171:
  for v,e,w being object, G1 being addEdge of G2,v,e,w
  st v <> w & G2 is c-tcolorable holds G1 is (c+`2)-tcolorable
proof
  let v,e,w be object, G1 be addEdge of G2,v,e,w;
  assume A1: v <> w & G2 is c-tcolorable;
  per cases;
  suppose A2: v is Vertex of G2 & w is Vertex of G2 & not e in the_Edges_of G2;
    consider t being TColoring of G2 such that
      A3: t is proper & card((rng t_V)\/rng t_E) c= c by A1;
    set V = rng t_V, E = rng t_E;
    set x = V \/ E, y = {x}, f = t_V +* (v .--> x), g = t_E +* (e .--> y);
    reconsider t1 = [f, g] as TColoring of G1 by A2, Th142;
    A4: {x,y} misses V \/ E & x <> y
    proof
      A5: not x in V \/ E;
      not y in V \/ E by TARSKI:def 1;
      hence {x,y} misses V \/ E by A5, ZFMISC_1:51;
      x in y by TARSKI:def 1;
      hence thesis;
    end;
    t1_V = t_V +* (v .--> x) & t1_E = t_E +* (e .--> y);
    then A6: t1 is proper by A1, A2, A3, A4, Th156;
    rng(e .--> y) = rng{[e,y]} by FUNCT_4:82
      .= {y} by RELAT_1:9;
    then A7: rng g c= E \/ {y} by FUNCT_4:17;
    rng(v .--> x) = rng{[v,x]} by FUNCT_4:82
      .= {x} by RELAT_1:9;
    then rng f c= V \/ {x} by FUNCT_4:17;
    then A8: rng f \/ rng g c= (V \/ {x}) \/ (E \/ {y}) by A7, XBOOLE_1:13;
    A9: (V \/ {x}) \/ (E \/ {y}) = V \/ ({x} \/ (E \/ {y})) by XBOOLE_1:4
      .= V \/ (({x} \/ {y}) \/ E) by XBOOLE_1:4
      .= (V \/ E) \/ ({x} \/ {y}) by XBOOLE_1:4
      .= (V \/ E) \/ {x,y} by ENUMSET1:1;
    card((V \/ E) \/ {x,y}) = card(V \/ E) +` card {x,y} by A4, CARD_2:35
      .= card(V \/ E) +` 2 by A4, CARD_2:57;
    then A10: card((V \/ E) \/ {x,y}) c= c +` 2 by A3, CARD_2:84;
    card(rng f \/ rng g) c= card((V \/ E) \/ {x,y}) by A8, A9, CARD_1:11;
    hence thesis by A6, A10, XBOOLE_1:1;
  end;
  suppose not(v is Vertex of G2 & w is Vertex of G2 &
      not e in the_Edges_of G2);
    then G1 == G2 by GLIB_006:def 11;
    then G1 is c-tcolorable by A1, Th167;
    hence thesis by Th161, CARD_2:94;
  end;
end;
